require(ggplot2)
require(lattice) # nicer scatter plots
require(plyr)
require(grid) # contains the arrow function
require(biOps)
require(doMC) # for parallel code
require(EBImage)
## To install EBImage
# source("http://bioconductor.org/biocLite.R")
# biocLite("EBImage")
# start parallel environment
registerDoMC()
# functions for converting images back and forth
im.to.df<-function(in.img) {
out.im<-expand.grid(x=1:nrow(in.img),y=1:ncol(in.img))
out.im$val<-as.vector(in.img)
out.im
}
df.to.im<-function(in.df,val.col="val",inv=F) {
in.vals<-in.df[[val.col]]
if(class(in.vals[1])=="logical") in.vals<-as.integer(in.vals*255)
if(inv) in.vals<-255-in.vals
out.mat<-matrix(in.vals,nrow=length(unique(in.df$x)),byrow=F)
attr(out.mat,"type")<-"grey"
out.mat
}
ddply.cutcols<-function(...,cols=1) {
# run standard ddply command
cur.table<-ddply(...)
cutlabel.fixer<-function(oVal) {
sapply(oVal,function(x) {
cnv<-as.character(x)
mean(as.numeric(strsplit(substr(cnv,2,nchar(cnv)-1),",")[[1]]))
})
}
cutname.fixer<-function(c.str) {
s.str<-strsplit(c.str,"(",fixed=T)[[1]]
t.str<-strsplit(paste(s.str[c(2:length(s.str))],collapse="("),",")[[1]]
paste(t.str[c(1:length(t.str)-1)],collapse=",")
}
for(i in c(1:cols)) {
cur.table[,i]<-cutlabel.fixer(cur.table[,i])
names(cur.table)[i]<-cutname.fixer(names(cur.table)[i])
}
cur.table
}
colMeans.df<-function(x,...) as.data.frame(t(colMeans(x,...)))
author: Kevin Mader date: 20 March 2014 width: 1440 height: 900 css: ../template.css transition: rotate
ETHZ: 227-0966-00L
We have dramatically simplified our data, but there is still too much.
56GB / sample
\[\downarrow\]
(1.75GB / sample)
cell.im<-jpeg::readJPEG("ext-figures/Cell_Colony.jpg")
cell.lab.df<-im.to.df(bwlabel(cell.im<.6))
size.histogram<-ddply(subset(cell.lab.df,val>0),.(val),function(c.label) data.frame(count=nrow(c.label)))
keep.vals<-subset(size.histogram,count>25)
cur.cell.id<-subset(cell.lab.df,val %in% keep.vals$val)$val[1]
cur.cell.df<-subset(cell.lab.df,val==cur.cell.id)
ggplot(subset(cell.lab.df,val %in% keep.vals$val),aes(x=x,y=y,fill=as.numeric(as.factor(val))))+
geom_raster()+
geom_tile(data=cur.cell.df,fill="red",alpha=0.5)+
labs(fill="Label",title="Larger Cells (>25px)")+
theme_bw(20)+coord_equal()
\[ I_{id}(x,y) = \begin{cases} 1, & L(x,y) = id \\ 0, & \text{otherwise} \end{cases}\]
mean.df<-colMeans.df(cur.cell.df[,c("x","y")])
ggplot(cur.cell.df,aes(x=x,y=y))+
geom_tile(color="black",fill="grey")+
geom_point(data=mean.df,color="red",pch=3,size=20)+
labs(title="Single Cell")+
theme_bw(20)+coord_equal()+guides(fill=F)
\[ \bar{x} = \frac{1}{N} \sum_{\vec{v}\in I_{id}} \vec{v}\cdot\vec{i} \] \[ \bar{y} = \frac{1}{N} \sum_{\vec{v}\in I_{id}} \vec{v}\cdot\vec{j} \] \[ \bar{z} = \frac{1}{N} \sum_{\vec{v}\in I_{id}} \vec{v}\cdot\vec{k} \]
If the gray values are kept (or other meaningful ones are used), this can be seen as a weighted center of volume or center of mass (using \(I_{gy}\) to distinguish it from the labels)
commean.fun<-function(in.df) {
ddply(in.df,.(val), function(c.cell) {
weight.sum<-sum(c.cell$weight)
data.frame(xv=mean(c.cell$x),
yv=mean(c.cell$y),
xm=with(c.cell,sum(x*weight)/weight.sum),
ym=with(c.cell,sum(y*weight)/weight.sum)
)
})
}
cur.cell.df.weight<-cbind(cur.cell.df,weight=with(cur.cell.df,(x-17)^2+(y-9)^2))
commean.df<-commean.fun(cur.cell.df.weight)
ggplot(cur.cell.df.weight,aes(x=x,y=y))+
geom_tile(aes(fill=weight),color="black")+
geom_point(data=commean.df,aes(x=xv,y=yv,color="COV"),pch=16,size=20)+
geom_point(data=commean.df,aes(x=xm,y=ym,color="COM"),pch=16,size=20)+
labs(title="Single Cell",color="Center",fill="Igy")+
theme_bw(20)+coord_equal()#+guides(fill=T)
\[ \Sigma I_{gy} = \frac{1}{N} \sum_{\vec{v}\in I_{id}} I_{gy}(\vec{v}) \] \[ \bar{x} = \frac{1}{\Sigma I_{gy}} \sum_{\vec{v}\in I_{id}} (\vec{v}\cdot\vec{i}) I_{gy}(\vec{v}) \] \[ \bar{y} = \frac{1}{\Sigma I_{gy}} \sum_{\vec{v}\in I_{id}} (\vec{v}\cdot\vec{j}) I_{gy}(\vec{v}) \] \[ \bar{z} = \frac{1}{\Sigma I_{gy}} \sum_{\vec{v}\in I_{id}} (\vec{v}\cdot\vec{k}) I_{gy}(\vec{v}) \]
Exents or caliper lenghts are the size of the object in a given direction. Since the coordinates of our image our \(x\) and \(y\) the extents are calculated in these directions
# since the edge of the pixel is 0.5 away from the middle of the pixel
emin<-function(...) min(...)-0.5
emax<-function(...) max(...)+0.5
extents.fun<-function(in.df) {
ddply(in.df,.(val), function(c.cell) {
c.cell.mean<-colMeans.df(c.cell[,c("x","y")])
out.df<-cbind(c.cell.mean,data.frame(xmin=c(c.cell.mean$x,emin(c.cell$x)),
xmax=c(c.cell.mean$x,emax(c.cell$x)),
ymin=c(emin(c.cell$y),c.cell.mean$y),
ymax=c(emax(c.cell$y),c.cell.mean$y)))
})
}
cell.extents<-extents.fun(cur.cell.df)
ggplot(cur.cell.df,aes(x=x,y=y))+
geom_tile(color="black",fill="grey")+
geom_segment(data=cell.extents,aes(x=xmin,y=ymin, xend=xmax,yend=ymax),color="red",size=2,lineend="square")+
labs(title="Single Cell")+
theme_bw(20)+coord_equal()+guides(fill=F)
Define extents as the minimum and maximum values along the projection of the shape in each direction \[ \text{Ext}_x = \left\{ \forall \vec{v}\in I_{id}: max(\vec{v}\cdot\vec{i})-min(\vec{v}\cdot\vec{i}) \right\} \] \[ \text{Ext}_y = \left\{ \forall \vec{v}\in I_{id}: max(\vec{v}\cdot\vec{j})-min(\vec{v}\cdot\vec{j}) \right\} \] \[ \text{Ext}_z = \left\{ \forall \vec{}\in I_{id}: max(\vec{v}\cdot\vec{k})-min(\vec{v}\cdot\vec{k}) \right\} \]
By definition (New Oxford American): varying in magnitude according to the direction of measurement.
Due to its very vague definition, it can be mathematically characterized in many different very much unequal ways (in all cases 0 represents a sphere)
\[ Aiso1 = \frac{\text{Longest Side}}{\text{Shortest Side}} - 1 \]
\[ Aiso2 = \frac{\text{Longest Side}-\text{Shortest Side}}{\text{Longest Side}} \]
\[ Aiso3 = \frac{\text{Longest Side}}{\text{Average Side Length}} - 1 \]
\[ Aiso4 = \frac{\text{Longest Side}-\text{Shortest Side}}{\text{Average Side Length}} \]
\[ \cdots \rightarrow \text{ ad nauseum} \]
Let's take some sample objects
# test function for ellipse generation
# ggplot(ldply(seq(-pi,pi,length.out=100),function(th) create.ellipse.points(a=1,b=2,th.off=th,th.val=th)),aes(x=x,y=y))+geom_path()+facet_wrap(~th.val)+coord_equal()
create.ellipse.points<-function(x.off=0,y.off=0,a=1,b=NULL,th.off=0,th.max=2*pi,pts=36,...) {
if (is.null(b)) b<-a
th<-seq(0,th.max,length.out=pts)
data.frame(x=a*cos(th.off)*cos(th)+b*sin(th.off)*sin(th)+x.off,
y=-1*a*sin(th.off)*cos(th)+b*cos(th.off)*sin(th)+y.off,
id=as.factor(paste(x.off,y.off,a,b,th.off,pts,sep=":")),...)
}
deform.ellipse.draw<-function(c.box) {
create.ellipse.points(x.off=c.box$x[1],
y.off=c.box$y[1],
a=c.box$a[1],
b=c.box$b[1],
th.off=c.box$th[1],
col=c.box$col[1])
}
test.objs<-data.frame(x=0,y=0,th.off=0,a=5,b=rev(c(1e-3,1e-2,1e-1,1,2,3,4,5)))
test.objs$col<-1:nrow(test.objs)
test.objs$x.extents<-2*test.objs$a
test.objs$y.extents<-2*test.objs$b
ggplot(ddply(test.objs,.(col),deform.ellipse.draw),
aes(x=x,y=y,group=as.factor(id)))+
geom_polygon(color="black",fill="blue")+coord_equal()+
facet_grid(col~.)+theme_bw(15)
s.aiso<-function(lside,sside) (lside/sside-1)
aiso<-function(lside,sside) ((lside-sside)/lside)
a.aiso<-function(lside,sside) (2*lside/(lside+sside)-1)
as.aiso<-function(lside,sside) (2*(lside-sside)/(lside+sside))
awrap<-function(c.fun) function(in.ell.dim) c.fun(pmax(2*in.ell.dim$a,2*in.ell.dim$b),pmin(2*in.ell.dim$a,2*in.ell.dim$b))
aiso.table.fn<-function(in.objs) {
ddply(in.objs,.(b),function(c.ell) {
data.frame(Aiso1=awrap(s.aiso)(c.ell),
Aiso2=awrap(aiso)(c.ell),
Aiso3=awrap(a.aiso)(c.ell),
Aiso4=awrap(as.aiso)(c.ell))
})
}
aiso.table<-aiso.table.fn(test.objs)
names(aiso.table)[1]<-"Y Extent"
kable(aiso.table,digits=2)
| Y Extent | Aiso1 | Aiso2 | Aiso3 | Aiso4 |
|---|---|---|---|---|
| 0.00 | 4999.00 | 1.00 | 1.00 | 2.00 |
| 0.01 | 499.00 | 1.00 | 1.00 | 1.99 |
| 0.10 | 49.00 | 0.98 | 0.96 | 1.92 |
| 1.00 | 4.00 | 0.80 | 0.67 | 1.33 |
| 2.00 | 1.50 | 0.60 | 0.43 | 0.86 |
| 3.00 | 0.67 | 0.40 | 0.25 | 0.50 |
| 4.00 | 0.25 | 0.20 | 0.11 | 0.22 |
| 5.00 | 0.00 | 0.00 | 0.00 | 0.00 |
ggplot(ddply(test.objs,.(col),deform.ellipse.draw),
aes(x=x,y=y,group=as.factor(id)))+
geom_polygon(color="black",fill="blue")+coord_equal()+
facet_wrap(~col)+theme_bw(15)
names(aiso.table)[1]<-"y.extent" ggplot(aiso.table,aes(x=y.extent))+ geom_line(aes(y=Aiso1,color="Aiso1"))+ geom_line(aes(y=Aiso2,color="Aiso2"))+ geom_line(aes(y=Aiso3,color="Aiso3"))+ geom_line(aes(y=Aiso4,color="Aiso4"))+ scale_y_log10()+ labs(x="Y extents",y="Anisotropy",color="Metric")+ theme_bw(20)
Objects with uniformally distributed, independent \(x\) and \(y\) extents
m.count<-5000
many.objs<-data.frame(x=0,y=0,th.off=0,a=runif(m.count,0,20),b=runif(m.count,0,20))
many.objs$col<-1:nrow(many.objs)
many.objs$x.extents<-2*many.objs$a
many.objs$y.extents<-2*many.objs$b
maiso.table<-aiso.table.fn(many.objs)
ggplot(ddply(subset(many.objs,col<16),.(col),deform.ellipse.draw),
aes(x=x,y=y,group=as.factor(id)))+
geom_polygon(color="black",fill="blue")+coord_equal()+
facet_wrap(~col,ncol=5)+theme_bw(15)
ggplot(many.objs)+ geom_density(aes(x=x.extents,color="X"),size=2)+ geom_density(aes(x=y.extents,color="Y"),size=2)+ #scale_x_log10()+ labs(x="Extent Length (pixels)",color="Axis")+ guides(fill=F)+ xlim(0,80)+ theme_bw(20)
splom(subset(maiso.table,Aiso1<10)[,c("Aiso1","Aiso2","Aiso3","Aiso4")],pch=16)
ggplot(maiso.table)+ geom_density(aes(x=Aiso1,color="Aiso1"),size=2)+ geom_density(aes(x=Aiso2,color="Aiso2"),fill="green",alpha=0.5)+ geom_density(aes(x=Aiso3,color="Aiso3"),size=2)+ geom_density(aes(x=Aiso4,color="Aiso4"),size=2)+ #scale_x_log10()+ labs(x="Anisotropy",color="Metric")+ xlim(0,2)+guides(fill=F)+ theme_bw(20)
ggplot(subset(cell.lab.df,val %in% keep.vals$val),aes(x=x,y=y,
fill=as.numeric(as.factor(val))))+
geom_raster()+
geom_tile(data=cur.cell.df,fill="red",alpha=0.5)+
labs(fill="Label",title="Labeled Cells")+
theme_bw(20)#+coord_equal()
cell.extents.all<-extents.fun(subset(cell.lab.df,val %in% keep.vals$val)) ggplot(cell.extents.all,aes(x=x,y=y))+ geom_segment(aes(x=xmin,y=ymin, xend=xmax,yend=ymax),color="red",lineend="square")+ geom_point(aes(color=as.numeric(as.factor(val))),size=2)+ labs(title="COM and Extents",color="Label")+ theme_bw(20)#+coord_equal()
mean.pos.subfun<-function(in.df) {
ddply(in.df,.(val),function(c.cell) {
c.mean<-colMeans.df(c.cell)
data.frame(x=c.cell$x-c.mean$x,y=c.cell$y-c.mean$y)
})
}
mean.sub.df<-mean.pos.subfun(subset(cell.lab.df,val %in% keep.vals$val))
ggplot(cell.extents.all,aes(x=x-x,y=y-y))+
geom_raster(data=mean.sub.df,aes(x=x,y=y))+
geom_segment(aes(x=xmin-x,y=ymin-y, xend=xmax-x,yend=ymax-y),
color="red")+
geom_point(color="red",size=2)+
facet_wrap(~val,ncol=14)+
labs(x="x - COM",y="y - COM")+
theme_bw(15)+coord_equal()
bbox.fun<-function(in.df) {
ddply(in.df,.(val), function(c.cell) {
c.cell.mean<-colMeans.df(c.cell[,c("x","y")])
xmn<-emin(c.cell$x)
xmx<-emax(c.cell$x)
ymn<-emin(c.cell$y)
ymx<-emax(c.cell$y)
out.df<-cbind(c.cell.mean,
data.frame(xi=c(xmn,xmn,xmx,xmx,xmn),
yi=c(ymn,ymx,ymx,ymn,ymn),
xw=xmx-xmn,
yw=ymx-ymn
))
})
}
cell.bbox.all<-bbox.fun(subset(cell.lab.df,val %in% keep.vals$val))
ggplot(cell.bbox.all,aes(x=x-x,y=y-y))+
geom_raster(data=mean.sub.df,aes(x=x,y=y))+
geom_path(aes(x=xi-x,y=yi-y),
color="red")+
geom_point(color="red",size=2)+
facet_wrap(~val,ncol=14)+
labs(x="x - COM",y="y - COM")+
theme_bw(15)+coord_equal()
cell.bbox.all$aiso<-with(cell.bbox.all,(pmax(xw,yw)-pmin(xw,yw))/pmax(xw,yw)+val/1e5)
ggplot(cell.bbox.all,aes(x=x-x,y=y-y))+
#geom_raster(data=mean.sub.df,aes(x=x,y=y))+
geom_path(aes(x=xi-x,y=yi-y,group=val),
color="red")+
geom_point(color="red",size=2)+
facet_wrap(~aiso,ncol=14)+
labs(x="x - COM",y="y - COM")+
theme_bw(15)+coord_equal()
While easy to calculate, the bounding box / extents approach is a very rough approximation for most of the objects in our image. In particular objects which are not parallel to the \(XY\)-axes are misrepresented.
While many of the topics covered in Linear Algebra and Statistics courses might not seem very applicable to real problems at first glance, at least a few of them come in handy for dealing distributions of pixels (they will only be briefly covered, for more detailed review look at some of the suggested material)
Similar to K-Means insofar as we start with a series of points in a vector space and want to condense the information. With PCA instead of searching for distinct groups, we try to find a linear combination of components which best explain the variance in the system.
As an example we will use a very simple example of corn and chicken prices vs time
n.pts<-20
test.pca<-data.frame(time=c(1:n.pts),
corn.price=0.01*runif(n.pts)+0.05)
test.pca$chicken.price<-with(test.pca,corn.price+0.01*runif(n.pts))
ggplot(test.pca,aes(x=time))+
geom_line(aes(y=corn.price,color="Corn Price ($/kg)"))+
geom_line(aes(y=chicken.price,color="Chicken Price ($/g)"))+
labs(x="Time (days)",y="Price",color="Commodity")+
theme_bw(20)
test.pca.princomp<-princomp(test.pca[,-1]) test.pca$pca1<-test.pca.princomp$scores[,1] test.pca$pca2<-test.pca.princomp$scores[,2] ggplot(test.pca,aes(x=time))+ geom_line(aes(y=pca1,color="Principal Component #1"))+ geom_line(aes(y=pca2,color="Principal Component #2"))+ geom_line(aes(y=corn.price,color="Corn Price ($/kg)"))+ geom_line(aes(y=chicken.price,color="Chicken Price ($/g)"))+ labs(x="Time (days)",y="Price",color="Commodity")+ theme_bw(20)
The first principal component condenses the correlated information in both the chicken and corn prices (perhaps the underlying cost of fuel) since it explains the most variance in the final table of corn and chicken prices.
ggplot(test.pca,aes(x=pca1))+ geom_point(aes(y=corn.price,color="Corn Price ($/kg)"))+ geom_point(aes(y=chicken.price,color="Chicken Price ($/g)"))+ labs(x="Principal Component #1",y="Price",color="Commodity")+ theme_bw(20)
The second principal component is then related to the unique information seperating chicken from corn prices but neither indices directly themselves (maybe the cost of antibiotics)
ggplot(test.pca,aes(x=pca2))+ geom_point(aes(y=corn.price,color="Corn Price ($/kg)"))+ geom_point(aes(y=chicken.price,color="Chicken Price ($/g)"))+ geom_point(aes(y=5*(chicken.price-corn.price),color="Difference in Chicken-Corn"))+ labs(x="Principal Component #2",y="Price",color="Commodity")+ theme_bw(20)
Going back to a single cell, we have the a distribution of \(x\) and \(y\) values.
A principal component analysis of the voxel positions, will calculate two new principal components (the components themselves are the relationships between the input variables and the scores are the final values.)
pca.fun<-function(in.df) {
ddply(in.df,.(val), function(c.cell) {
c.cell.cov<-cov(c.cell[,c("x","y")])
c.cell.eigen<-eigen(c.cell.cov)
c.cell.mean<-colMeans.df(c.cell[,c("x","y")])
out.df<-cbind(c.cell.mean,
data.frame(vx=c.cell.eigen$vectors[1,],
vy=c.cell.eigen$vectors[2,],
vw=sqrt(c.cell.eigen$values),
th.off=atan2(c.cell.eigen$vectors[2,],c.cell.eigen$vectors[1,]))
)
})
}
cur.cell.id<-subset(cell.lab.df,val %in% keep.vals$val)$val[100]
cur.cell.id<-58
cur.cell.df<-subset(cell.lab.df,val==cur.cell.id)
cell.pca<-pca.fun(cur.cell.df)
ggplot(cur.cell.df,aes(x=x,y=y))+
geom_tile(color="black",fill="grey")+
geom_segment(data=cell.pca,aes(x=x,y=y,xend=x+vx*vw,yend=y+vy*vw,
color=as.factor(round(vw*10)/10)),
arrow=arrow(length = unit(0.3,"cm")),size=2)+
labs(title="Single Cell",color="Score /\nEigenvalue")+
theme_bw(20)+coord_equal()+guides(fill=F)
We start off by calculating the covariance matrix from the list of \(x\), \(y\), and \(z\) points that make up our object of interest.
\[ COV(I_{id}) = \frac{1}{N} \sum_{\forall\vec{v}\in I_{id}} \begin{bmatrix} \vec{v}_x\vec{v}_x & \vec{v}_x\vec{v}_y & \vec{v}_x\vec{v}_z\\ \vec{v}_y\vec{v}_x & \vec{v}_y\vec{v}_y & \vec{v}_y\vec{v}_z\\ \vec{v}_z\vec{v}_x & \vec{v}_z\vec{v}_y & \vec{v}_z\vec{v}_z \end{bmatrix} \]
We then take the eigentransform of this array to obtain the eigenvectors (principal components, \(\vec{\Lambda}_{1\cdots 3}\)) and eigenvalues (scores, \(\lambda_{1\cdots 3}\))
\[ COV(I_{id}) \rightarrow \begin{bmatrix} \vec{\Lambda}_{1x} & \vec{\Lambda}_{1y} & \vec{\Lambda}_{1z} \\ \vec{\Lambda}_{2x} & \vec{\Lambda}_{2y} & \vec{\Lambda}_{2z} \\ \vec{\Lambda}_{3x} & \vec{\Lambda}_{3y} & \vec{\Lambda}_{3z} \end{bmatrix} * \begin{bmatrix} \lambda_1 \\ \lambda_2 \\ \lambda_3 \end{bmatrix} \] The principal components tell us about the orientation of the object and the scores tell us about the corresponding magnitude (or length) in that direction.
princomp or pca in various languages and scale well to very large datasets.While the eigenvalues and eigenvectors are in their own right useful
vec.to.ellipse<-function(pca.df) {
ddply(pca.df,.(val),function(cur.pca) {
# assume there are two vectors now
create.ellipse.points(x.off=cur.pca[1,"x"],y.off=cur.pca[1,"y"],
b=sqrt(5)*cur.pca[1,"vw"],a=sqrt(5)*cur.pca[2,"vw"],
th.off=pi/2-atan2(cur.pca[1,"vy"],cur.pca[1,"vx"]),
x.cent=cur.pca[1,"x"],y.cent=cur.pca[1,"y"])
})
}
cell.ellipse<-vec.to.ellipse(cell.pca)
ggplot(cur.cell.df,aes(x=x,y=y))+
geom_tile(color="black",fill="grey")+
geom_segment(data=cell.pca,aes(x=x,y=y,xend=x+vx*vw,yend=y+vy*vw,
color=as.factor(round(vw*10)/10)),
arrow=arrow(length = unit(0.3,"cm")),size=2)+
geom_path(data=cell.ellipse)+
labs(title="Single Cell",color="Score /\nEigenvalue")+
theme_bw(20)+coord_equal()+guides(fill=F)
ggplot(cur.cell.df,aes(x=x,y=y))+ geom_tile(color="black",fill="grey")+ geom_path(data=bbox.fun(cur.cell.df),aes(x=xi,y=yi,color="Bounding\nBox"))+ geom_path(data=cell.ellipse,aes(color="Ellipse"))+ labs(title="Single Cell",color="Shape\nAnalysis\nMethod")+ theme_bw(20)+coord_equal()+guides(fill=F)
cell.pca.all<-pca.fun(subset(cell.lab.df,val %in% keep.vals$val))
cell.ell.all<-vec.to.ellipse(cell.pca.all)
ggplot(cell.bbox.all,aes(x=x-x,y=y-y))+
geom_raster(data=mean.sub.df,aes(x=x,y=y))+
#geom_path(aes(x=xi-x,y=yi-y),color="red")+
geom_path(data=cell.ell.all,aes(x=x-x.cent,y=y-y.cent),color="red")+
geom_point(color="red",size=2)+
geom_segment(data=cell.pca.all,aes(xend=vx*vw,yend=vy*vw),
arrow=arrow(length = unit(0.3,"cm")),color="red")+
facet_wrap(~val,ncol=14)+
labs(x="x - COM",y="y - COM")+
theme_bw(15)+coord_equal()
pca.aiso.all<-ddply(cell.pca.all,.(val),function(cur.pca) {
lon.ax<-max(cur.pca$vw)
sho.ax<-min(cur.pca$vw)
data.frame(e.aiso=(lon.ax-sho.ax)/lon.ax)
})
both.aiso<-merge(pca.aiso.all,cell.bbox.all,by="val")
no.aiso.vals<-subset(both.aiso,aiso<0.01 & e.aiso>0.2)$val
ggplot(both.aiso,aes(x=aiso,y=e.aiso))+
geom_point(aes(color=ifelse(aiso<0.01,"lost information","normal")))+geom_smooth()+
labs(x="Extents / BB Anisotropy",y="Elliptical Anisotropy",color="Points")+
coord_equal()+theme_bw(20)
We see that there seems to be a general, albeit weak, correlation between the two measures. The most concerning portion is however the left side where the extents or bounding box method reports 0 anisotropy and the elliptical method reports substancial amounts of it.
ggplot(subset(cell.bbox.all,val %in% no.aiso.vals),aes(x=x-x,y=y-y))+
geom_raster(data=subset(mean.sub.df,val %in% no.aiso.vals),aes(x=x,y=y))+
geom_path(aes(x=xi-x,y=yi-y,color="BB"))+
geom_path(data=subset(cell.ell.all,val %in% no.aiso.vals),
aes(x=x-x.cent,y=y-y.cent,color="Ellipse"))+
geom_point(color="red",size=2)+
geom_segment(data=subset(cell.pca.all,val %in% no.aiso.vals),
aes(xend=vx*vw,yend=vy*vw),
arrow=arrow(length = unit(0.3,"cm")),color="red")+
facet_wrap(~val)+
labs(x="x - COM",y="y - COM")+
theme_bw(15)+coord_equal()
The models we have done are all applicable to both 2D and 3D images. The primary difference is when looking at 3D images there is an extra dimension to consider.
xg<-seq(-10,10,length.out=41)
make.ellipse.3d<-function(a,b,c,xg) {
ell.3d<-expand.grid(x=xg,y=xg,z=xg)
ell.3d$val<-with(ell.3d,(x/a)^2+(y/b)^2+(z/c)^2 < 1)
ell.3d
}
cloud(z~x*y,data=subset(make.ellipse.3d(9,3,10,xg),val),xlim=c(-10,10),ylim=c(-10,10),zlim=c(-10,10))
It can also be shown as a series of 2D slices.
ggplot(subset(make.ellipse.3d(9,3,10,xg),val),aes(x=x,y=y))+ geom_tile(color="blue")+ facet_wrap(~z)+ coord_equal()+ xlim(c(-10,10))+ylim(c(-10,10))+ theme_bw(10)
Anisotropy applies to these samples as well but it only gives us information about the shortest and the longest dimensions. Is that enough?
Each of these images has the exact same anisotropy because the shortest semiaxis length remains
xg<-seq(-10,10,length.out=41) b.vals<-seq(3,10,length.out=4) cloud(z~x*y | as.factor(b), data=ldply(b.vals,function(b) cbind(subset(make.ellipse.3d(b,10,3,xg),val),b=b)), xlim=c(-10,10),ylim=c(-10,10),zlim=c(-10,10))
If we take a slice through the image we can image the final shape looking like a very small thin rod in the case where the second dimension is equal to 3 (short in two directions and long in the other) and a pankcake (short in one direction and long in the other two).
ggplot(ldply(rev(b.vals),function(b) cbind(subset(make.ellipse.3d(b,10,3,xg),val & z==0),b=b)), aes(x=x,y=y))+ geom_tile(aes(fill=as.factor(round(b*10)/10)),alpha=1)+ facet_wrap(~z)+ coord_equal()+ xlim(c(-10,10))+ylim(c(-10,10))+ labs(title="Profile Cut-through of Ellipse",fill="Second\nDimension\nLength")+ theme_bw(20)
We can thus introduce a new metric for assessing the second-degree anisotropy in the object and this we shall somewhat more formally call Oblateness.
\[ \textrm{Ob} = 2\frac{\lambda_{2}-\lambda_{1}}{\lambda_{3}-\lambda_{1}}-1 \]
The value like in anisotropy is bound between -1 and 1.
ggplot(ldply(rev(b.vals),function(b) cbind(subset(make.ellipse.3d(b,10,3,xg),val & z==0),b=b)), aes(x=x,y=y))+ geom_tile(aes(fill=as.factor(round(b*10)/10)),alpha=1)+ facet_wrap(~z)+ coord_equal()+ xlim(c(-10,10))+ylim(c(-10,10))+ labs(title="Profile Cut-through of Ellipse",fill="Second\nDimension\nLength")+ theme_bw(20)
ob.vals<-data.frame(b=b.vals,obl=2*(b.vals-3)/(10-3)-1
)
ggplot(ob.vals,aes(x=b,y=obl))+
geom_line()+
labs(x="Second Dimension Length",y="Oblateness")+
theme_bw(20)
Many physical and chemical processes occur at surfaces and interfaces and consequently these structures are important in material science and biology. For this lecture surface and interface will be used interchangebly and refers to the boundary between two different materials (calcified bone and soft tissue, magma and water, liquid and gas) Through segmentation we have identified the unique phases in the sample under investigation.
Constructing a mesh for an image provides very different information than the image data itself. Most crucially this comes when looking at physical processes like deformation.
While the images are helpful for visualizing we rarely have models for quantifying how difficult it is to turn a pixel off
base.grid<-expand.grid(x=c(-10:10),y=c(-10:10))
base.shape.undef<-cbind(base.grid,val=with(base.grid,abs(x)<6 & abs(y)<6))
def.grid<-rbind(
cbind(base.shape.undef,type="Before"),
cbind(base.grid,val=with(base.grid,abs(x)<10 & abs(y)<4),type="After")
)
ggplot(def.grid,
aes(x=x,y=y,fill=ifelse(val,"on","off")))+
geom_tile(alpha=0.80,color="black")+
coord_equal()+facet_wrap(~type)+
xlim(c(-10,10))+ylim(c(-10,10))+
labs(title="Pixel-based Deformation",fill="Value")+
theme_bw(20)
If the image is turned into a mesh we now have a list of vertices and edges. For these vertices and edges we can define forces. For example when looking at stress-strain relationships in mechanics using Hooke's Model \[ \vec{F}=k (\vec{x}-\vec{x_0}) \] the force needed to stretch one of these edges is proportional to how far it is stretched.
df.to.mesh<-function(in.df,d.size.x=0.5,d.size.y=0.5,make.seg=F) {
corner.pts<-expand.grid(dx=c(-d.size.x,d.size.x),dy=c(-d.size.y,d.size.y))
min.x<-min(in.df$x)
max.x<-max(in.df$x)
min.y<-min(in.df$y)
max.y<-max(in.df$y)
mesh.pts<-ddply(in.df,.(x,y),function(cur.df) {
xy.cols<-c("x","y")
pos.data<-cur.df[1,xy.cols]
n.data<-cur.df[,!(names(cur.df) %in% xy.cols)]
ddply(corner.pts,.(dx,dy),function(c.offset) {
cbind(x=pos.data$x+c.offset$dx[1],y=pos.data$y+c.offset$dy[1],n.data)
})
})
if(make.seg) {
grid.pts<-expand.grid(dx=c(-d.size.x,0,d.size.x),dy=c(-d.size.y,0,d.size.y))
grid.pts$seg.len<-with(grid.pts,sqrt(dx^2+dy^2))
subset(
ddply(subset(grid.pts,((abs(dx)==0) + (abs(dy)==0))==1),.(dx,dy,seg.len),
function(c.offset) {
cbind(xend=mesh.pts$x+c.offset$dx[1],yend=mesh.pts$y+c.offset$dy[1],
mesh.pts)
}),
xendmin.x & yendmin.y)
} else {
mesh.pts
}
}
undef.mesh<-df.to.mesh(subset(base.shape.undef,val),make.seg=T)
base.shape.def<-base.shape.undef
base.shape.def$x<-base.shape.def$x*10/6
base.shape.def$y<-base.shape.def$y*4/6
def.mesh<-df.to.mesh(subset(base.shape.def,val),
d.size.x=0.5*10/6,d.size.y=0.5*4/6,make.seg=T)
ggplot(rbind(cbind(undef.mesh,type="Before"),
cbind(def.mesh,type="After")),
aes(x=x,y=y))+
geom_segment(aes(xend=xend,yend=yend,color=sign(seg.len-0.5)),size=2)+
geom_point(size=2,color="green",pch=20)+
coord_equal()+facet_wrap(~type)+
scale_color_gradient2(mid="black",low="red",high="blue",space="Lab")+
#xlim(c(-10,10))+ylim(c(-10,10))+
labs(title="Mesh-based Deformation",color="(x-x0)")+
theme_bw(20)
Since we uses voxels to image and identify the volume we can use the voxels themselves as an approimation for the surface of the structure.
From this we can create a mesh by
A wide variety of methods of which we will only graze the surface (http://en.wikipedia.org/wiki/Image-based_meshing)
Voxels are very poor approximations for the surface and are very rough (they are either normal to the x, y, or z axis and nothing between). Because of their inherently orthogonal surface normals, any analysis which utilizes the surface normal to calculate another value (growth, curvature, etc) is going to be very inaccurate at best and very wrong at worst.
The image is processed one voxel at a time and the neighborhood (not quite the same is the morphological definition) is checked at every voxel. From this configuration of values, faces are added to the mesh to incorporate the simplist surface which would explain the values.
Marching tetrahedra is for some applications a better suited approach
So while bounding box and ellipse-based models are useful for many object and cells, they do a very poor job with the sample below.
cur.cell.df<-subset(cell.lab.df,val==155) cell.pca<-pca.fun(cur.cell.df) cell.ellipse<-vec.to.ellipse(cell.pca) ggplot(cur.cell.df,aes(x=x,y=y))+ geom_tile(color="black",fill="grey")+ geom_path(data=bbox.fun(cur.cell.df),aes(x=xi,y=yi,color="Bounding\nBox"))+ geom_path(data=cell.ellipse,aes(color="Ellipse"))+ labs(title="Single Cell",color="Shape\nAnalysis\nMethod")+ theme_bw(20)+coord_equal()+guides(fill=F)
